Photographic method for making geographic maps



p 1953 V G E A FALK 2,650,517

PHOTOGRAPHIC METHOD FOR MAKING GEOGRAPHIC MAP Sept 1, 1953v G. E. A.FALK 2 5559 517 I PHOTOGRAPHIC METHCD FOR MAKING GEOGRAPHIC MAPS- FiledFeb. 10, 1949 ll Sheets-$heet 2 lA/lfA/ZOA Gem-mgr) EA FAl-k p 1, 1953G. E. A. FALK 2,650,517

PHOTOGRAPHIC METHOD FOR MAKING GEOGRAPHIC MAPS Filed Feb. 10, 1949 llSheets-Sheet 5 PU F If M lA/kE/VI'OP GERflARD EA. PAL/q Sept. 1, 1953 G.E. A. FALK 2,650,517

I PHOTOGRAPHIC METHOD FOR MAKING GEOGRAPHIC MAPS Filed Feb. 10, 1949 11Sheets-Sheet 6 Fig.9

Sept. 1, 1953 G. E. A. FALK 2,650,517

PHOTOGRAPHIC METHOD FOR MAKING GEOGRAPHIC MAPS Filed Feb. 10, 1949 11Sheets-Sheet 7 IIVVf/W'UP GERHARD A- HM-K Sept. 1, 1953 G. E. A. FALK2,650,517

PHOTOGRAPHIC METHOD FOR MAKING GEOGRAPHIC MAPS Filed Feb. 10, 1949 11Sheets-Sheet 8 lyjja' 11b 'INVENMP:

GERHRRD EA. FALK P 1953 ca. E. A. FALK 2,650,517

PHOTOGRAPHIC METHOD FOR MAKING GEOGRAPHIC MAPS Filed Feb. 10, 1949 11Sheets-Sheet 9 5 7 IIIIIIIIII Ill, 6 3* f 5 G-ERHARD E'..A. FALK Sept.1, 1953 G. E. A. FALK 2,650,517

PHOTOGRAPHIC METHOD FOR MAKING GEOGRAPHIC MAPS Filed Feb. 10, 1949 llSheets-$heet 10 INVEWTUP QERHARD E. PAL-K Sept. 1, 1953 I G. E. A. FALK2,650,517

PHOTOGRAPHIC METHOD FOR MAKING GEOGRAPHIC MAPS Filed Feb. 10, 1949 11Sheets-Sheet 11 Patented Sept. -1, 1953 UNITED STATES PATENT OFFICEPHOTOGRAPHIC METHOD FOR MAKING GEOGRAPHIC MAPS Gerhard Ernst AlbrechtFalk, Hamburg, Germany 7 Claims.

The invention relates to a geographical map, and more particularly butnot exclusively to a town map, and to methods of making such maps.

In many cases localities, especially towns, consist of a denselybuilt-up town centre, which merges gradually into suburbs, which aremuch less densely built-up. In order to be able to represent thestreets, buildings, etc., on a town map, one is therefore forced to showlocalities in a relatively large scale, generally 1:10.000 to 120,000,although the less densely built-up districts outside the centre of thetown would permit the use of a smaller scale without affecting the goodlegibility of the marking of the streets, etc. The maps of suchlocalities are therefore generally unhandy and difficult to use; thefrequent unfolding and folding up of a large map causes considerablewear. The division of a map into sepa rate sheets, which are boundtogether in the form of a book, on the other hand, makes the mapconsiderably more difficult to read.

The purpose of the invention is to eliminate these disadvantages.According to the invention, certain parts of the locality areillustrated on a relatively large scale, whereas the other parts of thelocality are shown on scales which continuously decrease from the chosenmaximum scale.

A map constructed according to the invention is preferably made by adistorted photograph of a map having an unvarying scale.

In general it will be sufiicient and preferable if the distorted scalevalues have only one maximum. In this case the geometrical locus of thegreatest scale is a point or a straight line.

The method of producing a map in which the geometrical locus of thegreatest scale is a straight line comprises the steps of placing a maphaving an invariable scale parallel to a light sensitive surface, and ofbending the map about a straight line in a plane parallel to said lightsensitive surface so as to form two portions of the map which arearranged symmetrically to a plane normal to is light sensitive surfaceand passes through said straight line whereupon the map is projected ona light sensitive surface to produce a distorted photograph of the map.

In order to produce a map having a point as geometrical locus of thegreatest scale, a distorted photograph showing a distorted map andproduced by the above method is placed in a plane which is parallel toanother light sensitive surface. Then the distorted photograph is bentout of the plane about a straight line which extends at an angle to thestraight line representing the geometrical locus of the distorted map.The dis- Y 2 torted map is then photographed on the other lightsensitive surface whereby a map is produced which is distorted in twodirections and has a point as geometrical locus.

The above mentioned photographs are preferably made in several steps, inthat at each step the map is placed on an evolvable surface, such as acylindrical surface and the photograph obtained thereby is placed on afurther evolvable surface, at an angle to the first, and this photographserves as the basis for the next step. It is generally suliicient if thephotographs are made in two consecutive steps by means of two similarevolvable surfaces placed at right angles to each other.

In order to produce a geographical map, in which the geometrical locusof the largest scale is a point, the evolvable surfaces are preferablycylinders of which the cross-sections are conic sections, the verticesof which, if they are present, are so arranged that their distances fromthe point of projection are less than that of the other points of theconic section. If, in this case, the course of the conic section leadsfrom a certain region to a too great diminution of the scale, thenaccording also to the invention a tangent is drawn to the conic sectionfrom this region.

Instead of cylinders of which the cross-sections are conic sections,prisms having cross-sections in the form of polygons, of which the onecorner is so arranged that its distance from the point of projection isless than that of the other corners, may be provided. Preferably atriangle is used as the polygon.

In order to produce approximately a geographical map, in which the locusof the greatest scale is a straight line, the map with the invariablescale is so placed on a prism that the points of the straight line inwhich the maximum scale is situated, lie at a smaller distance from thepoint of projection than the other points of the individual prismcross-sections. In a photographic projection the distance of the pointof projection from the individual points of that straight line, whichshould be the locus of the greatest scale, is variable. In consequencethe scale actually decreases along this straight line as the distancesof individual points in this straight line from the point of projectionincrease. Here, too, conic sections, conic sections with tangentsthereto, or polygons can be used as cross-sections for the prisms.

The accompanying drawing illustrate the invention, by way of example. Inthe drawings:

Figure 1 is a view of a lattice network of a town Figure 2 is adiagrammatic illustration of a projection carried out according to theinvention;

Figure 3 is a lattice network which has been similarly distorted by themethod illustrated in Figure 2;

Figure. 4 is. a diagrammatic illustration of a n. other projectioncarried out according to the invention;

Figure 5 is a lattice network which has been similarly distorted by themethod illustrated; in Figure 4;

Figure 6 is a diagrammatic illustratio of; a further projection carriedout according to the invention;

Figure 7 is a lattice network which has been similarly distorted by themethod illustrated Figure 6;

Figure 8 is a e ialsrs iis er iesen ei en x a n ho orthog nal he proje in. at e" arrie out ph ilgm hic ly;

"Figure 9 isan illustration of a square net- WWK. n mcqss i-9 5 9ml". newo k;

Figure lo. illustration showing the geometrical solution of atransformation of the squar'e networkof Figure 9 made according tocertainlaws;

Figures lla -c are an illustration of a method of determininggeometrically the cross-section of the projection cylinder and the.degre network. for thefirst stpin the photographic pro-- duction of' asquare-network transformed according to Figure 10;

Figures 1200 0 are. an illustration of rnethod oidetermininggeometrically the, trflssrsection of thefprojection cylinder. and thedegre net-:. work off the, second step, in the photographic productionof asquare-network transformed ac cording to Figurev 10';

Figure 13. is"'an. illustration of a, square-nets work "superimposed onan ellipse, sector network;

'Ffigu'rfe" lil is. an illustration showin the geometrical solution of;a v transformation of the. square-network I of Figure. 13 1 carried out.accord? ing' to certain laws;

Figure 15. is an illustration of a square-net work; which issuperimposed in .part on acircle sector. network andin part on anellipse sector newer-1;, three. ellips'es.oi different signs. being ureis. is. an illustration showing. the, geemetrical solution ofa"transiorrnation of the.

In theexample boththe. distances; ZS=Z are.

dividedintosix equal parts. which aremarked consecutively withtheinumloers. l to In Figure 2 itis assumed th.at S Z;l gradually d cr ss to =..3/? l.- brre pwdin ly TZ=m should decrease to T'Z=m=3/ lm. Inthis. emplars?.merwii m ad s network having a triangular cross-section,the one half of which is represented in the upper part I of Figure 2 bythe triangular area ZSPI. The map is placed in such a way that the lineTZT of Fig. 1 appears as a point Z in Figure 2.

The map of Figure 1 is then photographed through a lens 0, and the imagethus obtained, the axis TT of which has not been shortened, is placed onthe face 21 of a prism. also of triangular cross-section, the one halfof which is represented in the lower part II of Figure 2 by thetriangular area Z'IP; In this case the map must be so arranged that lineSZS which has been shortened by the preceding photographical rar iesi onis ers ndic r to the plane of the drawing. The photograph, which is nowtaken gives a lattice network having the desired reduction of scale inwhich m is continuously shortened to 3/4 m=m' and l to 3/4 2:2.

The selection of the point of projection 0 depends on the focal lengthof F of the objective lens of the camera used for the photograph. Its.distance from the map centre Z, which lies v on the vertices of the twoprisms which face the point 0 and have the above described triangularcross-sections, the halves of which. are equal. to, the triangles ZSP1and. ZTPz, is,2 F,if the photorgraph is to be in natural size. Thenearer the. point of projection moves to the prisms nsing, r a aa W d ragl Qbi t r h reater. s, t e e i e. st tionf f he l i e net work, that.is the. nearer the outer, ed es ofv the ma move towards he man centre n.the pro:- jection.

In Figure 2 Z5 and. Z1. are sections through. the plane of the map andZS. and Z 'I. aresec io s thr u the p e ti n. pla e Usi he symbols shownin Figure 2, the-following equations are obtained and, according to thesine law,

tice networkobtained-by;thismethod. It will be;

seen from thisthat decreaseof the. scale starts, in the map centre Z andproceedsuniformly. to the map edges. have a visible kink or bendin the-twomap axes;

In Figure 3 the. scale decreases frozn 1:11,300;in.

the centre of the mapto 1 13,900: atits edges.

The pyramid projection is especially suitable for thecarto-graphicalrepresentation of'towns, the. densely built-up, area .ofvwhich is ,only, of. small extent.

line, thelattice network of Figure 1 needs onlyv to be pl c on a in l rsmhav ns a trian u ar. cross-section and 'proj ected only once.

In another method of carrying-out theinven-v tion the lattice networkofFigure ,1 is placed ,for.

The; lattice lines thus produced If the densely built.-up,,p,arts of,the town extend more or less along. a straight,

the first photographic projection on a cylinder I (Figure 4) having theradius 11, which is so calculated that, for example, Z=3/4Z again. Forthe second photograph the map is placed on the cylinder II, the radiusT2 of which is so calculated that for example m=3/4m again. This methodof carrying out the invention is hereinafter called bi-cylinderprojection. The same distance of the projection point from the mapcentre OZ=2F is used as basis for the projection as in the case ofpyramid projection.

If it be assumed that in Figure 4 the length of the chord ZS is equal tothe arc ZS=Z, we obtain for the approximate calculation of the radii ofthe two cylinders 2 cos a 2 2 cos 01;;

Similarly as described in connection with Figure 2 tan 5:5

2F sin B I sin 6:

Figure shows a lattice network transformed by means of a bi-cylinderprojection, the scale of which is in the centre at Z equal to 110,000and at the edges 1116,000. Since the map is square and not rectangularthe two cylinders used in the bi-cylinder project-ion have equal radii.

The decrease of the scale is only very small in the vicinity of the mapcentre in this projection, but all the greater at the map edges. It istherefore especially well suited for the cartographical representationof a town having an extensive, densely built-up centre and scatteredbuilt-up outlying districts.

In a further form of carrying out the invention the map of Figure l, isas shown in Figure 6, first placed for photographing on a body I, whichis composed of a cylinder segment with relatively small radius T1 and aplane tangentially touching the cylinder segment. After this aphotograph is taken on a similarly shaped body II with the radius T2.The amount of distortion is dependent on the size of the radius of thecylinder segment and the position of the line along which the tangentplane touches the cylinder segments (that is the positions of P1 and P2in the cross-section of Figure 6). The amount of the distortion may beselected, for example, so that Z'=3 41 with reference to the body I andthat m'=3/4m with reference to body 11. In Figure 6,

M1P1=r1=radius of the projection cylinder I,

P1S=section through the tangential'plane on the projection cylinder I,

M2Pz==r2=radius of the projection cylinder II,

P2T=section through the tangential plane on the projection cylinder II.

Figure 7 shows a lattice network, which has been transformed in themanner explained with reference to Figure 6. Owing to the cross-sectionof the body used in this transformation this method of carrying out theinvention is hereinafter called cylinder tangent projection. It will beseen from Figure '7 that the cylinder tangent projection is intermediatebetween the pyramid projection of Figure 3 and the bi-cylinderprojection of Figure 5. Whilst in the bi-cylinder projection of Figure 5the scale decreases only very slightly in a relatively large circlearound the map centre, this region of an only very slight scale decreaseis considerably smaller in the cylinder tangent projection of Figure 7.In the region of the tangential plane the scale decrease increasesuniformly up to the map edges. The cylinder tangent projection istherefore suitable especially for the representation of localities withsmall inner built-up areas and relatively slightly spread out outerdistricts.

All projections can be used not only if the densely built-up centre ofthe town lies in the centre of the map, but can be carried out for allasymmetrical cases, in which, therefore, the densely built-up centre ofthe town, for the representation or" which a larger scale is desirable,lies outside the centre of the map.

Furthermore, ellipsoids, paraboloids and hyperbloiols can be used asprojection bodies. The selection of the projection body, of itsdimensions and of the distance of the point of projection from theprojection body depends in all cases on the character of the town to bepictured and on the desired size of map.

For the sake of completeness it may be pointed out that other scales areused as basis for the lattice network of Figures 3, 5 and '7 than forthe drawings of Figures 1, 2 4 and 6.

It will often be advantageous to project the map to be transformed froma globe or sphere, since this projection is easy and simple to calculatefor all sizes of maps. Figure 8 shows how an orthogonal globe projectionmay be carried out photographically. In this figure a part of a spherewith the radius 11, which is shown in section is projected orthogonallyonto a plane E which is tangent to the sphere. In this projection thearch of the sphere S1Z=Z is distorted into SZ l. This distortionincreases as the angle w increases, for which reason this angle can becalled the angle of distortion. The projection of an arc is alwaysZ'=r1. sin w. The quotient q= Z/Z' is called the quotient of distortionand is constant for a certain angle w. It can be determined from a tablefor o==0 to 99. 0n the basis of Figure 8, the following equations areobtained:

In order to be able to carry out the orthogonal projection of Figure 8photographically, two alterations must be made:

(1) Since the map to be distorted cannot be 7 placed on a sphere, it isplaced on a cylinder 7 with a radius of curvature r1 equal to that ofthe sphere and shortened by double photographical projection, in thatthe map is projected once with the north-south direction vertical andthe second time with the northsouth direction, horizontal by which meansthe effect of a globe projection is obtained;

(2) Since no parallel projection can be made photographically, the pointof projection O must be displaced from the infinite to the finite. Forthis purpose a projection body of the cross-section SZPI is constructedon to which the map to be distorted of the half length 82:2 is placedand when photographed the same distortion SZ=Z' results as theorthogonal projection of the cylinder section SrZMr produces.

In order to obtain the curve SZ of the desired projection body, thepoint of projection O .is on the one hand connected to the points ofintersection of the lattice lines of the projected map 8'2; and theseprojection lines are prolonged. beyond the projection plane E. In theexample, the map is divided into six equal parts i to 8 so that, if theprojection lines OZ and OS runprojection beam at a point, which is apoint in the cross-section line S2 of the desired projection body.Around this point an arc of a circle is again struck with the sameradius 1/6, the point of intersection of which with the prolongation ofthe next projection beam gives a second point of the curve. Thecontinuation of this process finally gives the whole line 82, whichrepresents a cross-section line through the .desired projection bodySZPr. The greater the number or" the lattice squares, the smaller istheir size, and therefore the greater is the accuracy of the method. Themap to be distorted is placed on this projection body and shortened byphotographic double projection as above described.

If a map with an orthogonal lattice network (Figure l) is to be soshortened that the size of the map is to be 25% smaller, the suitableangle must be found from the above-described table giving the quotientof distortion for q=Z/Z.. For the reduction q= l/3= l.3, the angle w=70.According to the above equation 4, by inserting for 2 half the length ofthe map and 70 for the value of w, one obtains the radius T1 of thecylinder on to which the map must be placed, in order to obtain inparallel projection the desired reduction of the size by distortion ofthe scale. If the distance of the top edge of the map from the centre ofthe map represents 6 km. on the ground, that is six lattice squares witha side 1/6 Z, the points of intersection of the lattice lines are foundon the projection plane E, in that the individual values for l' arecalculated according to the above Equation 2 for the values w=70/6,w=2.'70/6 etc. If the number of lattice squares is equal to n the scaleof the edge can be calculated from the side of the last square. Bycomparison with the scale of the centre of the .map, which isapproximately equal to the scale of the original, one can "thenascertain whether the degree of distortion can be used for the desiredpurpose.

In this way the radius of the cylinder can be calculated :forasymmetrical cases, in which the densely built-up centre of the towndoes not lie in the centre of the map, in that, while maintaining theangle of distortion and thus the same scale at the edge, the distance ofthe centre of the town from the edge of the map is inserted in the aboveEquation 4 .for Z and thus for each map quadrant the radius of thecylinder section is calculated, and then the points of intersection ofthe lattice lines in the plane of projection are calculated inaccordance with Equation 2 and after that the projection body isconstructed.

The sphere projection is, of course, very similar in its character tothe bl-cylinder projection described with reference to Figures 4 and 5,of which it represents merely a special case. For q:1r/2 one obtains,for example, with a scale in the centre of the map of l:l0,000 a scaledistor tion at the edges of l:ll0,000, which in practice could scarcelybe used.

The network of squares shown in Figure 9, for example, is to be sotransformed that o r carrying out the transformation the scale i edgesof the network is half as large as in the centre. The alteration of thepale should pro ceed continuously from the centre and be the same in alldirections. It is also necessary that the alteration of the scale is inproportion to a power of the distance from the centre point. This meanswith the references of Figure 9 that on the line 1 of the square networkthe following relations hold good:

In order to solve this problem geometrically the square network is, asshown in superimposed on a circle sector-network, wherein the number ofcircles is equal to .12 and the distances between two consecutivecircles constant. The sectors also always include equal angles.

The circle network is new transformed in the required way. Whole circleslie the square network. A number x has to be found such that IBM-1%,that is to say If the distance apart of the circles is C, then after thetransformation it .is 0411', where k is the order number of the relevantcircle. Hence the radii of the-circles are case it does not matterwhetherin the observation the image (on the plane of projection E) andthe object lie on one side of the projecting point or are separated byit. The following con sideration refers to the first case.

The radial straight line shortening is given by the precedingconsiderations, that by the fact that the straight lines are as follows:

ghrhi:hitils:ik:kl etc.

In the following only the hatched square at the top right hand of thewhole lattice square network of Figure 9 is projected.

Figure 11 is given for explaining a method by which the cross-section ofthe projection cylinder and the degree network is determined for thefirst step in the photo-geometrical production of a square networktransformed according to Figure 10. In the left half 12a of this figurethe cross-section of the hyperboloid is represented by ABC, on thehyperboloid surface AB of which the hatched part of Figure 9 is soplaced that the straight line 9 lies at right angles to the plane of thedrawing. Instead of this one could also start with the straight line 'ilying at right angles to the plane of the drawing. A cross-sectionthrough the projection surface is represented by the line E, on which,starting from the point B, the radial distances shortened in accordancewith the above given law are plotted.

Whereas in the Figures 9 and 10 the radial distance is subdividedseventeen times, the last parts are omitted in Figure 11. Now theindividual section points on the line E are con nected to the projectionpoint 0 and extended beyond the line E. Now a circle is drawn around Bwith a radius, which is equal to the undistorted unit section of aradial ray. This circular arc meets the first projection ray at a pointon the hyperbolic surface AB of the cross-section of the hyperboloidABC. A circle is now drawn with the radius around this point, whichmeets the next projection ray in a further point of the curve AB. Thecontinuation of this process finally gives the whole course of the curveAB and thus the crosssection of the hyperboloid ABC. The right-hand part1 lb of Figure 11 shows a view of the diagram of Figure 11 viewed in thedirection of the arrow. The straight lines 72, i, k, Z, m and n are alsomarked on the curve AB, n coinciding with the point A.

From Figure 11b one obtains the projection of the square network, whichis shown in 110 and with it is ended the first step of the method.

Each point of intersection in the square network.

of Figure 9 corresponds to a point of intersection of the square networkof Fig. 11b. The individual points of intersection are connected to theprojection point 0 of Figure 111) in exactly the same manner as theprojections of these points of intersection on the curve AB are connected to the projection point. The individual points of the projectedimage of Figure 110 are given by the points of intersection of these projection rays with the projection plane. By way of explanation, any pointX in Figure 110 is assumed which is indicated in Figure 1112 by Xb andin Figure 11a by Xa The point X in Figure 110 is given by the line (1 ofFigure 1112 and the line e of Figure 11a. The other points of thelattice network of Figure 110 are found in corrasponding manner.

The lattice network of Figure 110 is placed on the same cylinder withthe hyperbolic surface AB in such a way that the line 1 lies in thepoint B 10 perpendicularly to plane of the drawing. A projection fromthe projection point 0 on to the projection plane E is now made of thelattice network thus formed. In this way the final transformed latticenetwork of Figure 120 is obtained, which agrees with the transformedlattice net- Work shown in Figure 10. Graphically the lattice network ofFigure 120 may be obtained from Figures 12a and 12b in the same way asthe lattice network of Figure was obtained. from Figures 11a and 111).For the determination of Figure the method is only somewhat morecomplicated, since the lines I to 6 are no longer parrallel to eachother and to the line 1. Owing to this in Fig. 12h, the points ofintersection of the individual lines l, 2, 3, 4, 5, 6 and 1 with thelines g to n are projected in individual points, in the exampletherefore in seven points, whereas these points of intersection inFigure 11a are projected only in one point each. 1

By similar principles it is possible to make a photographic projectionfor lattice networks, which are to be transformed according to any lawswhatever.

Figure 13 shows square network which has been superimposed on an ellipsesector network. Figure 14 shows the square network transformed accordingto the invention, the transformation having been carried out accordingto the same laws as that of Figure 10 and Figure 120.

Figure 15 shows a similar square network, which has been superimposed onfour sectors each of 90. The sector at the right-hand bottom corner is asector of a circle, whereas the other three sections are ellipsesectors, the ellipses of which have parameters of different sizes.

Figure 16 shows the transformed square network, the transformationhaving been carried out according to the same laws as those of Figures10, 12c and 14.

I claim:

1. A method for producing a geographical map having a scale graduallydecreasing in all directions, comprising in combination, the steps ofplacing a fiat map having an invariable scale with a straight line inthe plane thereof parallel to a light sensitive surface; bending the mapabout said straight line out of said plane parallel to said lightsensitive surface and away from the same into two portions arrangedsymmetrically to a plane passing through said straight line normal tosaid lightsensitive surface, and projecting the map on said lightsensitive surface whereby a distorted reproduction is produced showingthe map at a scale gradually decreasing in two opposite directionsnormal to and away from said line; placing a map produced by the abovemethod with another straight line in the plane thereof and extending atan angle to said firstmentioned straight line and crossing the sameparallel to another light sensitive surface; bending said map producedby the above method about said other straight line out of said planeparallel to said other light sensitive surface and away from the sameinto two portions arranged symmetrically to a plane passing through saidother straight line normal to said other light sensitive surface; andprojecting said map produced by said above method on said other lightsensitive surface whereby a distorted reproduction is produced showingthe map at the scale gradually decreasing from the point of intersectionbetween said straight line and said other straight line.

assume 2. A method for producing a geographical map having a scalegradually decreasing in all directions, comprising in combination, thesteps oi placing a flat map having an invariable scale with straightline in the plane thereof parallei to a light sensitive surface; bendingthe map about said straight line out :of said plane parallel to saidlight sensitive surface and away from the same into two straightportions arranged symmetrically to a plane passing through said straightline normal to said light sensitive surface, and projecting the on saidlight sensi tive surface whereby a distorted reproduction producedshowing the map at a scale gradually decreasing in two oppositedirections normal to and away'from line; placing a map produced by theabove method with another straight line in the plane thereofandeX-tending at an angle to said firstementioned straight line :andcrossing the same parallel to another light sensitive surface; bendingmap produced by the above method. about said other straight line out ofsaid plane el to said other light sensitive surface and away the sameinto two straight portions arranged symmetrically to a plane passingthrough said other straight .line normal to said other light sensitivesurface; and projecting said map produced by said above method on saidother light sensitive surface whereby a distorted reproduction isproduced showing the map at the scale gradually decreasing from thepoint of tersect on between said straight line and other straight line.

3. .A method for producing a geographical map having a scalegraduallydecreasing in all directions, comprising in combination, thesteps of placing a flat map having an invariable scale with a straightline in the plane thereof parallel to a light sensitive surface; bendingthe map about said straight line out of said plane parallel to saidlight sensitive surface and away from the same into two arcuate portionsarranged. symmetrically to a plane passing through said straight linenormal to said light sensitive surface, and projecting the map on saidlight sensi" tive surface whereby a distorted reproduction is producedshowing the map at a scale gradually decreasing in two oppositedirections normal to and away from said line; placing a map produced bythe above method with another straight line in the plane thereof andextending at an angle to said first-mentioned straight line and crossingthe same parallel to another light sensitive surface; bending said mapproduced by the above method about said other straight line out of saidplane parallel to said other light sensitive surface and away from thesame into two arcuate portions arranged symmetrically to a plane passingthrough said other straight line normal to said other light sensitivesurface; and projecting said map produced by said above method on saidother light sensitive surface whereby a distorted reproduction isproduced showing the map at the scale gradually decreasing from thepoint of intersection between said straight line and said other straightline.

l. A method for producing a geographical map having a scale graduallydecreasing in all directions, comprising in combination, the steps ofplacing a flat map having an invariable scale with a straight line inthe plane thereof parallel to a light sensitive surface; bending the mapabout said straight line out of said plane parallel to said lightsensitive surface and away from the same into two portions having theshape of a conic sectionarranged symmetrically to a plane passingthrough said straight line normal to said light sensitive surface, andprojecting the map on said light sensitive surface-whereby a distortedreproduction is produced showing the map at a scale gradually-decreasingin two opposite directions normal to and away from said line; placing amap produced by the above method with another straight line in the planethereof and extending at an angle to said first-mentioned. straightvline and crossing the same parallel to another light sensitive surface;bending said map produced by the above method about said other straightline out of said plane parallel to other light sensitive :surface :andaway from the same into two portions having the shape of conic sectionarranged symmetrically to a plane passing through said other straightline normal to said otherflight sensitive surface; and projecting saidmap produced by said above method on said other light sensitive surfacewhereby a distorted reproduction is produced showing the map at thescale gradually decreasing from the point of intersection between saidstraight line and said other straight line.

5. A method for producing l'argeographical map having a scale graduallydecreasing in all directions, comprising in combination, the steps ofplacing a fiat map having an invariable scaie with a straight line inthe plane thereof parai-- lel to a light sensitive surface; bending theabout said straight line out of said plane par T lel to saidlight-sensitive surface and away from the same into two portions havingthe shape of a conic section with the vertex thereof placed in saidstraight line arranged -symmetrically to a plane passing through .saidstraight line norcoal to said light sensitive surface, and projectingthe map on said light sensitive surface where by a distortedreproduction is produced shov ing the map at a scale graduallydecreasing in two opposite directions normal to and away from said line;placing a map produced by the ab o method with another straight line inthe p thereof and extending at an angle to said hi. mentioned straightline and crossing the same parallel to another light sensitive surface;bend" ing said map pro'duced'by the above method about said otherstraight line-out of said plane parailei to said other light sensitivesurface and away from the same into two portions having the shape of aconic section with the vertex thereof placed in said other straight'line arranged symmetricah ly to a plane passing through said otherstraight line normal to said other light sensitive surface; andprojecting said map produced by said above method on said other lightsensitive surface whereby a distorted reproduction is produced showingthe map at the scale gradually decreasing from the point of intersectionbetween said straight line and. said other straight line.

A method for producing a geographical map having a scale graduallydecreasing in all direc tions, comprising in combination, the steps ofplacing a first map having an invariable scale with a straight line inthe plane thereof parallel to a light sensitive surface; bending the mapabout said straight line out of said plane parallel to said lightsensitive surface away from the came into two portions having the shapeof a conic section with the vertex thereof placed said straight line andending in a tangent thereof arranged symmetrically to .a plane passingthrough said straight line normal to said light sensitive surface, andprojecting the map on said light sensitive surface whereby a distortedreproduction is produced showing the map at a scale gradually decreasingin two opposite directions normal to and away from said line; placing amap produced by the above method with another straight line in the planethereof and extending at an angle to said first-mentioned straight lineand crossing the same parallel to another light sensitive surface;bending said map produced by the above method about said other straightline out of said plane parallel to said other light sensitive surfaceand away from the same into two portions having the shape of a conicalsection with the vertex thereof placed in said other straight line andending in a tangent thereof arranged symmetrically to a plane passingthrough said other straight line normal to said other light sensitivesurface; and projecting said map produced by said above method on saidother light sensitive surface whereby a distorted reproduction isproduced showing the map at the scale gradually decreasing from thepoint of intersection between said straight line and said other straightline.

7. A method for producing a geographical map having a scale graduallydecreasing in two directions, comprising in combination, the steps ofplacing a flat map having an invariable scale with a straight line inthe plane thereof parallel to a light sensitive surface; bending the mapabout said straight line out of said plane parallel to said sensitivesurface and away from the same into two portions arranged symmetricallyto a plane passing through said straight line normal to said lightsensitive surface, and projecting the map on said light sensitivesurface whereby a distorted reproduction is produced showing the map ata scale gradually decreasing in two opposite directions normal to andaway from said line; placing a map produced by the above method withanother straight line in the plane thereof and extending at an angle ofto said first-mentioned straight line and crossing the same parallel toanother light sensitive surface; bending said map produced by the abovemethod about said other straight line out of said ,plane parallel tosaid other light sensitive surface and away from the same into twoportions arranged symmetrically to a plane passing through said otherstraight line normal to said other light sensitive surface; andprojecting said map produced by said above method on said other lightsensitive surface whereby a distorted reproduction is produced showingthe map at the scale gradually decreasing from the point of intersectionbetween said straight line and said other straight line.

GERHARD ERNST ALBRECHT FALK.

References Cited in the file of this patent UNITED STATES PATENTS NumberName Date 521,064 Weyde June 5, 1894 1,456,954 Lucken May 29, 19231,528,021 Janzer Mar. 3, 1925

